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Brezis-Kamin type results involving locally integrable weights

2023; Khayyam Publishing; Volume: 37; Issue: 1/2 Linguagem: Inglês

10.57262/die037-0102-99

0893-4983

Ailton Rodrigues da Silva, Diego Ferraz, Pedro Ubilla,

Differential Equations and Boundary Problems

We study existence and asymptotic behavior of entire positive bounded solutions for the following class of semilinear elliptic problem \begin{equation*} \left\{ \begin{aligned} L(u) & = \varrho(x) f(x,u) & \text{ in } & \mathbb{R}^N,\ N \geq 3,\\ u & > 0, & \text{ in } & \mathbb{R}^N, \end{aligned} \right. \end{equation*} where $0 \leq \varrho \in L^p_{loc}(\mathbb{R}^N),$ for some $ N < p \leq \infty.$ Here, $L$ is a local uniform elliptic operator and $f(x,s)$ is a nonlinearity with sublinear behavior at zero and at $+\infty$. This type of result has already been studied in the celebrated work by H. Brezis and S. Kamin for the case when $L= -\Delta$ and $\varrho \in L^{\infty}_{loc}(\mathbb{R}^N)$. Our approach allows us to include for instance $ - \text{dive} \left ( (1+|x|^\mu )^\nu\nabla u \right ) = u^q(|x|^\alpha + |x| ^\beta)^{-1}$ with suitable $\alpha,\ \beta > 0,$ $\mu,\ \nu \in \mathbb{R}$ and $0 < q < 1.$ Here, we include two local uniform elliptic situations: $\mu > 0$ with $\nu =1$ or $\nu =-1.$